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Suppose you decide to play roulette 20 times. Each time, you bet$30 on BLACK, $20 on ODD and $10 on the 5-NUMBER bet. What is yourexpected total gain?...

Question

Suppose you decide to play roulette 20 times. Each time, you bet$30 on BLACK, $20 on ODD and $10 on the 5-NUMBER bet. What is yourexpected total gain?

Suppose you decide to play roulette 20 times. Each time, you bet $30 on BLACK, $20 on ODD and $10 on the 5-NUMBER bet. What is your expected total gain?



Answers

Find the expected payback for the games of chance described in Exercises 47–52.
Roulette In one form of roulette, you bet $\$ 1$ on "even." If 1 of the 18 even numbers comes up, you get your dollar back, plus another one. If 1 of the 20 noneven $(18 \text { odd, } 0, \text { and } 00 \text { ) }$ numbers comes up, you lose your dollar.

Karen is probably playing a game with a deck of cards and they were wanting to find the expected value for this team. Now generally of X. Is equal to the sum of all the excess times the probability of each facts, people, the sum of X times the probability of lecture. So we need to take each situation separately. Now a person is paid $15 for a jack or queen And so we win $15 for jack or a queen. There's a total of 52 cards in our deck And there are a total of eight jacks or queens. So this is acts times probability backs 15 times 8/52. Now we made $5 for drawing a king or an ace. That means we make $5 for kinase. There are eight kings and aces Out of a total of 52 and 20. And then to Epstein's of probability events. Yeah. Now a person who has any other card pays $4. So this means that we're gonna lose $4. That's in a uniform, We've already taken into account 16 cards. And so that means there's 36, we haven't taken into account, So there's 36 ways to lose. And so this is how we're going to find are expected value. There is by adding up these to get And when we do that, that gives us 4/13 for all of this year. And so expected value is four over $13.

The deer numbers probably give using the winning numbers in 20 years and the the office. Pretty fair. Dolo. There's a $1 bet you get back that corner futures in building number 37 years and the payoff will be minus one daughter because one another. But you don't get back. Get your news expected. Feel 1 38 35 minus by 10 years. I broke, you know, you know, fight to $6. You lose. You know, when your favorites

The question you are is about the winning in the road games. So you've been given them. There are chances off anyone off eating blacks lowers. So, like, we have 18 black slurs. Okay, then we have, Ah, 18 railed slaughters and, uh, do greens laws in the relatively. So therefore, I concede that the sample space story will be equal to 18 less 18 less to you guessed that the addition here gives me Will do 38 now moving further. Uh, we say that there are also two players, one bets on red and the other bets $1 in black. So the billing chance has joined you into the chance off to in 30 years for bold a player's trouble. So now I'm going to start with the initial part where I'm talking about winning the probably D for the Leo. Okay, Now you say that winning probability in bedding $1 all black that totally started it. So this would be 18 out off idiot right now, losing their for the losing probability for the scene will be equal to, uh, one minus eating by 30 years. Service old as you get 20 by the deed now to find the expectation off this probability. So we write on the formal of you affects, you know, it's given by submission off X into the probabilities. Okay, so then, in this case, we can write down this as one as a provided you're running into the probability of winning is eating by 20 year last minus one is the amount that he will lose so into the probability of losing is 30 by a solid 20 by 30 years. So if you simplify getting the expectation out here as minus 0.5 go six. So this you found the expectation off bedding $1 on black. So we can therefore, what you are saying that, uh though I think that is 5.26 person off a pro or Axum approximately say 5% chance if you don't print off the answer to 5% off losing. If you play 100 games by betting on black in this game, get now moving borders do for the simplify. This has a winning probability for the house now, so I will not talk about this. So now we can see that the House has a chance off the inning. And that probability is 26. Bite or deed. Okay, so we can say that. Therefore, the chance off losing right, I was it'll be one minus 36 buying 30 years if your soldiers you get grew by 28. Okay, so which is again, like, approximately 0.5 to 6. So we say that therefore, the house with me, because you don't find 05 to 6 times 100 play money approximately which is equal to again, say five person or I'm considering that I say $5 here. Okay, Now we need to find the standard deviation also for this so standard deviation. Oh, sure. Uh, it's actually going to be for the 100 roulette place or end in this case becomes 100. So I can easily find the standard edl now. So the standard error, the formula we apply is equal to all square root off in, which is a number off draws error. And this part when I'm writing yours becomes a standard delusions. It's the fraction off the winning into the fractions off losing, right. So if you multiply the whole Barbara getting this as 2.233 So now he's on. This answer is what I've got. I can therefore I have the final answer, for he has one less for B. He said that the house will make fine dollars bye giving or taking approximately $2. So that is dollar FIEs. But the less with a plus or minus off. $2. Okay. And then the answer. Be part we sent in the net gain for the house. Well, approximately be, say, $5 into you multiply by two. So we get this as $10 we said in the neck loss in the house will be approximately, say, $2 in tow to which gives evil who has $4 over here. Okay. And therefore I say that the two will players have of winning chance. There is, ah, dollar to lose or gain in each play in the long run

Here on this problem we're talking about expected by the expected value is the theoretical average. And we're dealing in American religion in roulette that we have 38 numbers. If a player bets $1 on a number of wins, the player keeps the dollar and receives an additional $35. Otherwise, the dollar is lost. So are expected value. Here there is a one out of 38 chance the person receives $35. And so there's a 37 out of 38 chance that the person loses the dollar entirely. And so when we evaluate this 35/38 minus 37/38 this gives us negative 0.5 And so on average, the person is expected to lose five cents.


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